Similar to non-local games (see [1]}), when the players can share a bipartite quantum state, a general strategy for a certificate game consists of a shared state \(|\Psi _{AB}\rangle  \in \mathcal {H}_A \otimes \mathcal {H}_B\)  between the two players, and two families of projective measurements \(M_A = \lbrace M_A(x)\rbrace _ {x \in f^{-1}(0)}\)  and \(M_B = \lbrace M_B(x)\rbrace _ {x \in f^{-1}(1)}\)  made on their respective part of the shared state. Here \(\mathcal {H}_A\)  and \(\mathcal {H}_B\)  are the Hilbert spaces of respective players. For each measurement \(M_*(x)\) , we denote the family of orthogonal projections as \(\lbrace P_{A;x,i}\rbrace _{i\in [n]}\)  (see [2]} for a definition of projective measurements).
